Conflict Resolution in Arbitrary Hypergraphs

We study conflict resolution algorithms in a multiple-access system in which there are n stations, and at any time only members of specified subsets of stations might need to transmit simultaneously on the same channel. This corresponds to saying that there is a hypergraph F such that stations i(1),..., i(k) may attempt to transmit at the same time on the channel if and only if i(1),..., i(k) are contained in some member of F. To the best of our knowledge, in the literature there are no papers studying the conflict resolution problem under the conflict model considered in this paper. When F is the family of all possible subsets of up to a certain number d of stations in {1,..., n}, our conflict model corresponds to the classical conflict model. In this paper, we focus on non-adaptive algorithms, that is, algorithms that schedule all transmissions in advance so that all stations transmit according to a predetermined protocol. It is well known that there exists a correspondence between superimposed codes and non-adaptive algorithms for conflict resolution in the classical conflict model [18]. In order to design our algorithms for the problem considered in the present paper, we introduce new combinatorial structures that generalize the notions of strongly selective families [4,5,18] and classical selectors [8], and are parameterized by the particular hypergraph at hand.